Sine rule
- Olympiad · IOQM
The sine rule says that in any triangle, each side divided by the sine of the opposite angle gives the same number.
Name the triangle ABC. Write a for the side opposite angle A, b for the side opposite B, and c for the side opposite C. Then:
- a / sin A = b / sin B = c / sin C.
In Class 10, trigonometric ratios live inside a right triangle. The sine rule works in any triangle. It links the longest side to the largest angle, and the shortest side to the smallest.
Use it when you know two angles and one side. You can also use it when you know two sides and an angle opposite one of them.
A small example
In triangle ABC, angle A = 30°, angle B = 45° and side a = 4 cm. Find side b.
- By the sine rule, b = a × sin B / sin A.
- sin 45° = 1/√2 and sin 30° = 1/2.
- So b = 4 × (1/√2) ÷ (1/2) = 8/√2 = 4√2 cm.
The values of sin 30° and sin 45° come from the ratios of specific angles. Angle B is bigger than angle A, and b = 4√2 is about 5.66, bigger than a. That fits the rule. When you know two sides and the angle between them, use the cosine rule instead.
Where it fits
This goes beyond the Class 10 board syllabus. HBCSE's syllabus for the Mathematical Olympiad names it under plane geometry. The Mathematics Teachers' Association (India), MTA(I), conducts IOQM, the first stage of the Mathematical Olympiad Programme that HBCSE organises for NBHM.
Sources
Facts last checked against these sources on 30 September 2026.
- Syllabus for Mathematical Olympiad (Plane Geometry) · HBCSE, TIFR
- Mathematical Olympiad 2026-2027: stages of selection · HBCSE, TIFR
- Brochure: Mathematical Olympiads 2026-2027 · HBCSE, TIFR
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